Using the Inverse
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Using the Inverse FFT For Filtering, Transient Details,
and Resampling
by Howard A. Gaberson
Abstract
The paper discusses the Fast Fourier Transform (FFT) and Inverse Fourier Transform. Equations are provided. A point of the article is that the transform is a set of amplitudes and phases of complex sine waves that when added together form a continuous curve. When the curve is evaluated at the signal sampling instants, it exactly reproduces the signal. Three applications of the inverse Fourier Transform are discussed, 1) band-wise reconstruction which includes low pass, high pass and band pass filtering, 2) reconstruction by eliminating smaller or large harmonics and 3) reconstruction to accomplish resampling.
PREVIEW
“Introduction:
The Fourier Transform (FT) has more uses than I can imagine; what I do with it is only a tiny fraction of its usefulness, but it is so central to machinery diagnostics that I want to show you some different uses I have found that I think provide useful insight. Hidden periodicities are our game; that which is repeated or cyclic in our signal, and the characteristics of this cyclic portion of our signal can be a clue for diagnosing machine condition? “What is cyclic?” is a fabulous sales angle. Good pitch charlatans have been snowing us with it forever: “…I’ve gone back and investigated the price of oober goobers over the last couple of centuries and have noticed a cyclic pattern, repeated maybe 12 or 13 times. It is a wonderful opportunity to get rich if you can discern this complicated pattern, which I know all about. Now is the perfect time to invest, and if you’ll give me all your money, I’ll make you rich.”
HOW TO LOOK AT THE DFT:
Anyway, back to our Fast Fourier Transform (FFT ): DFT really; I don’t care what kind of ingenious algorithm you use to calculate it as long as you perform Eq. (1) and Eq. (2). I taught myself the DFT starting from the complex exponential form of the Fourier series as Newland [1] suggests. You plug, the value of the Xk from Eq. (1) into Eq. (2), carefully keeping track of the indices, and by using the algebra formula for the sum of a geometric series [2], you find that they have to exactly work. Equation (1) transforms a list of x’s into a list of complex X’s, and Eq. (2) exactly transforms the complex X’s back to the original x’s. Amazing! But notice, I have the 1/N where it belongs, not where it is usually placed.
To use Eq. (1) and Eq. (2) for vibration analysis there are some ground rules. The data list is sampled at sampling rate fs. The xn of Eq. (1) are samples from a signal which was accurately sampled and band limited to fs/2; this means its Fourier Transform is zero for all frequencies greater than half the sampling rate. There is an even number of N samples in the list.”
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